R0047-24304 · q-20261015-advance-20261113-r07-finite-flow-network-0029
finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-capacity-audit；資料={"vertices":["s3","a3","t3","u3"],"edges":[["s3","a3"],["a3","s3"],["a3","t3"],["t3","a3"]],"source":"s3","sink":"t3","capacities":[20,20,20,20],"flows":[-4,8,12,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。
G001 — finite loop-free directed network、逐edge capacity與逐internal-vertex conservation。操作=flow-capacity-audit；資料={"vertices":["s3","a3","t3","u3"],"edges":[["s3","a3"],["a3","s3"],["a3","t3"],["t3","a3"]],"source":"s3","sink":"t3","capacities":[20,20,20,20],"flows":[-4,8,12,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。
操作前
source roles = [flow-capacity-audit]
操作後
derived = [[source, sink, edge_capacity_flow_rows, capacity_rows, capacity_feasible, first_failed_edge_or_null, certificate]]
條件：無
本頁列完整來源欄序，不需JavaScript；gradient boundary依原mode選6／8／7欄，不補欄也不由答案猜mode。
規則：source_ordered_fields_without_JS · 依賴：G001
操作前
source roles = [[s3, a3, t3, u3], [[s3, a3], [a3, s3], [a3, t3], [t3, a3]], [20, 20, 20, 20], [-4, 8, 12, 4], s3, t3]
操作後
derived = [[a3, s3, t3, u3], [[[a3, s3], 20, 8], [[a3, t3], 20, 12], [[s3, a3], 20, -4], [[t3, a3], 20, 4]], 2, 是]
條件：無
先依原index配好edge／capacity／flow，再按來源English canonical(edge)序排；vertices用UTF16。反向邊是不同edge，zero容量與孤立點合法，負flow保留供容量判失敗，不修改。
規則：Flow_zip_source_edges_capacities_flows_BEFORE_order · 依賴：G001, S001
操作前
source roles = [0, [a3, s3], 8, 20]
操作後
derived = [是, 是, 是]
條件：無
每條有向邊都要0<=flow<=capacity；網路signature合格並不保證候選flow可行。
規則：Flow_EACH_edge_two_sided_capacity_obligation · 依賴：G001, S002
操作前
source roles = [1, [a3, t3], 12, 20]
操作後
derived = [是, 是, 是]
條件：無
每條有向邊都要0<=flow<=capacity；網路signature合格並不保證候選flow可行。
規則：Flow_EACH_edge_two_sided_capacity_obligation · 依賴：G001, S003
操作前
source roles = [2, [s3, a3], -4, 20]
操作後
derived = [否, 是, 否]
條件：無
每條有向邊都要0<=flow<=capacity；網路signature合格並不保證候選flow可行。
規則：Flow_EACH_edge_two_sided_capacity_obligation · 依賴：G001, S004
操作前
source roles = [3, [t3, a3], 4, 20]
操作後
derived = [是, 是, 是]
條件：無
每條有向邊都要0<=flow<=capacity；網路signature合格並不保證候選flow可行。
規則：Flow_EACH_edge_two_sided_capacity_obligation · 依賴：G001, S005
操作前
Flow ALL incoming at vertex a3 := sum(-4, 4)
操作後
Flow ALL incoming at vertex a3 = 0
條件：無
依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。
規則：exact_rational_arithmetic · 依賴：G001, S006
操作前
Flow ALL outgoing at vertex a3 := sum(8, 12)
操作後
Flow ALL outgoing at vertex a3 = 20
條件：無
依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。
規則：exact_rational_arithmetic · 依賴：G001, S007
操作前
source roles = [a3, [2, 3], [0, 1]]
操作後
derived = [0, 20, 否, 是]
條件：無
入邊／出邊各依完整清單加總；孤立點是0=0。守恆只要求internal vertices，源匯仍計完整balance以求net值。
規則：Flow_EACH_vertex_complete_directional_incidence · 依賴：G001, S008
操作前
Flow ALL incoming at vertex s3 := sum(8)
操作後
Flow ALL incoming at vertex s3 = 8
條件：無
依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。
規則：exact_rational_arithmetic · 依賴：G001, S009
操作前
Flow ALL outgoing at vertex s3 := sum(-4)
操作後
Flow ALL outgoing at vertex s3 = -4
條件：無
依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。
規則：exact_rational_arithmetic · 依賴：G001, S010
操作前
source roles = [s3, [0], [2]]
操作後
derived = [8, -4, 否, 否]
條件：無
入邊／出邊各依完整清單加總；孤立點是0=0。守恆只要求internal vertices，源匯仍計完整balance以求net值。
規則：Flow_EACH_vertex_complete_directional_incidence · 依賴：G001, S011
操作前
Flow ALL incoming at vertex t3 := sum(12)
操作後
Flow ALL incoming at vertex t3 = 12
條件：無
依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。
規則：exact_rational_arithmetic · 依賴：G001, S012
操作前
Flow ALL outgoing at vertex t3 := sum(4)
操作後
Flow ALL outgoing at vertex t3 = 4
條件：無
依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。
規則：exact_rational_arithmetic · 依賴：G001, S013
操作前
source roles = [t3, [1], [3]]
操作後
derived = [12, 4, 否, 否]
條件：無
入邊／出邊各依完整清單加總；孤立點是0=0。守恆只要求internal vertices，源匯仍計完整balance以求net值。
規則：Flow_EACH_vertex_complete_directional_incidence · 依賴：G001, S014
操作前
Flow ALL incoming at vertex u3 := sum()
操作後
Flow ALL incoming at vertex u3 = 0
條件：無
依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。
規則：exact_rational_arithmetic · 依賴：G001, S015
操作前
Flow ALL outgoing at vertex u3 := sum()
操作後
Flow ALL outgoing at vertex u3 = 0
條件：無
依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。
規則：exact_rational_arithmetic · 依賴：G001, S016
操作前
source roles = [u3, [], []]
操作後
derived = [0, 0, 是, 是]
條件：無
入邊／出邊各依完整清單加總；孤立點是0=0。守恆只要求internal vertices，源匯仍計完整balance以求net值。
規則：Flow_EACH_vertex_complete_directional_incidence · 依賴：G001, S017
操作前
Flow source NET outflow not gross outgoing := sub(-4, 8)
操作後
Flow source NET outflow not gross outgoing = -12
條件：無
依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。
規則：exact_rational_arithmetic · 依賴：G001, S018
操作前
Flow sink NET inflow not gross incoming := sub(12, 4)
操作後
Flow sink NET inflow not gross incoming = 8
條件：無
依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。
規則：exact_rational_arithmetic · 依賴：G001, S019
操作前
source roles = [s3, t3, [[[a3, s3], 8, 20, 是], [[a3, t3], 12, 20, 是], [[s3, a3], -4, 20, 否], [[t3, a3], 4, 20, 是]], [[a3, 0, 20, 否], [u3, 0, 0, 是]], -12, 8]
操作後
derived = [否, 否, 否, 否, 否]
條件：無
全vertex總in−out中每edge相消，故全部internal守恆必得兩net相等；反向不成立，internal imbalance可互抵。容量、逐點守恆、兩net值分別核，feasible不推出maximum/min-cut。最後No是不提供最優證書，不否定特例恰好最優。
規則：Flow_value_capacity_conservation_and_maximum_separate · 依賴：G001, S020
S021 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S020 · premise_and_result_audit — 以精確有理數重算 Flow sink NET inflow not gross incoming，不以小數近似替代。S019 · premise_and_result_audit — 以精確有理數重算 Flow source NET outflow not gross outgoing，不以小數近似替代。S018 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S017 · premise_and_result_audit — 以精確有理數重算 Flow ALL outgoing at vertex u3，不以小數近似替代。S016 · premise_and_result_audit — 以精確有理數重算 Flow ALL incoming at vertex u3，不以小數近似替代。S015 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S014 · premise_and_result_audit — 以精確有理數重算 Flow ALL outgoing at vertex t3，不以小數近似替代。S013 · premise_and_result_audit — 以精確有理數重算 Flow ALL incoming at vertex t3，不以小數近似替代。S012 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S011 · premise_and_result_audit — 以精確有理數重算 Flow ALL outgoing at vertex s3，不以小數近似替代。S010 · premise_and_result_audit — 以精確有理數重算 Flow ALL incoming at vertex s3，不以小數近似替代。S009 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S008 · premise_and_result_audit — 以精確有理數重算 Flow ALL outgoing at vertex a3，不以小數近似替代。S007 · premise_and_result_audit — 以精確有理數重算 Flow ALL incoming at vertex a3，不以小數近似替代。S006 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S005 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S004 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S003 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。(s3, t3, [[[a3, s3], 20, 8], [[a3, t3], 20, 12], [[s3, a3], 20, -4], [[t3, a3], 20, 4]], [[[a3, s3], 8, 20, 是], [[a3, t3], 12, 20, 是], [[s3, a3], -4, 20, 否], [[t3, a3], 4, 20, 是]], 否, [[s3, a3], -4, 20, 否], every_edge_requires_zero_less_equal_flow_less_equal_capacity)
依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。
覆核日期：2026-10-26
原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 9c29c4ef812660675a46b8172a0f2493aacdf99d87eeabeb983b188f6b4958cccandidate record SHA-256: ad3d93c39da23836462e15cc97ea69642e529d2a80c6c55d99ba5dffcb37249esemantic review receipt SHA-256: 9b43b9823e9bd2478d6caa5b57ffed161c2c8956af86974e901d5662ad994386question-first witness SHA-256: eb97596068e2d5648033bab2b8f6ca9b653bbe8cb77c0f8e1d16d0c40e46f50bLean: NOT_RUN · human review: NOT_PERFORMED